Reliability and Risk Analysis. Survival and Reliability Function

Transcription

1 Reliability and Risk Analysis

2 Survival function We consider a non-negative random variable X which indicates the waiting time for the risk event (eg failure of the monitored equipment, etc.). The probability distribution of such a random variable X is in reliability theory usually described by the survival function S(x) S(x) = P(X > x), and using the distribution function of X we get S(x) = 1 F (x).

3 Survival function The survival function can be used to calculate the general moments of the random variable X. If exists kth general moment µ k = EX k, then µ k = k 0 x k 1 S(x)dx. (1) Furthermore, the survival function used to determine 100γ% lifetime T γ. 1 γ = F (x 1 γ) = F (T γ) = 1 S(T γ) which means that γ = S(T γ).

4 Example Suppose that a random variable X which has an exponential distribution Ex(λ), is the lifetime of some equipment. a) Find the survival function S(x) of the random variable X. b) Using S(x) calculate T 0,10; T 0,25; T 0,50; T 0,75 a T 0,90. c) Using S(x) calculate general moments µ 1, µ 2, µ 3, µ 4. d) Find the expected value EX, variance DX, skewness α 3 and kurtosis α 4. Express the required characteristics in general and then specify the values for λ = 0.1.

5 Example a) The distribution function F (x) is F (x) = 1 e λx, for x 0, the survival function is S(x) = 1 F (x) = e λx, x 0. b) From γ = S(T γ) = e λtγ we get 100γ% lifetime T γ, The required lifetimes are T γ = ln γ λ. T = λ T = λ T = λ T = λ T = λ = 23.03, = 13.86, = 6.93, = 2.88, = 1.05.

6 Example c) We can calculate general moments using (1). µ 1 = 1 0 x 0 e λx dx = [ 1 λ e λx ] 0 = 1 λ = 10. Other general moments can be calculated by the per-partes method. The general formula is of the form µ k = k!, for k = 1, 2, 3,.... λk µ 2 = 2 λ = 200, 2 µ3 = 6 24 = 6 000, µ4 = = λ3 λ4

7 Example d) the expected value EX = µ 1 = 1 λ = 10, the variance skewness DX = E(X EX ) 2 = EX 2 (EX ) 2 = 1 λ 2 = 100, E(X EX [ ] )3 α 3 = = λ 3 µ DX 3/2 3 3µ 2µ 1 + 2µ 3 1 = [ = λ 3 3! λ 3 2! ( ) ] λ 2 λ + 2 = = 2 λ

8 Example kurtosis E(X EX [ ] )4 α 4 = = λ 4 µ DX 2 4 4µ 3µ 1 + 6µ 2µ 2 1 3µ 4 1 = [ = λ 4 4! λ 4 3! 1 4 λ 3 λ + 6 2! ( ) 2 ( ) ] = 4! 4! = 9. λ 2 λ λ

9 When describing risk events, such as the description of the process failures, natural disasters etc., the probability distribution of the waiting time for such an event (we call waiting time for failure) is usually distribution with heavy tails. It is therefore important to investigate the behavior of tails distribution of time to failure. It can be done by so call the risk function s(x) which is defined as follows s(x) = f (x) 1 F (x) = f (x) S(x), where F the distribution function, S is the survival function f is the density of time to failure X. The risk function s(x) is not defined for x when F (x) = 1. The risk function s(x) is in reliability theory called the reliability function.

10 For a given time point x > 0 and small time interval x > 0 we will find the conditional probability that the reference element breaks down in the interval (x, x + x assuming that to the x it worked without disorders. If X is a trouble-free operation time of this element, we get P(x < X x + x X > x) = = P(x < X x + x) P(X > x) P(x < X x + x X > x) P(X > x) = = 1 F (x + x) F (x) x. S(x) x F (x + x) F (x) S(x) = =

11 F (x+ x) F (x) x If x is close to 0, then the difference is close derivative of the distribution functionf (x) (if the derivative in x exists). Since F (x) = f (x), we get for small values x the approximate relation P(x < X < x + x X > x) =. f (x) x = s(x) x. S(x) This relation shows that the risk function is local characteristics of reliability.

12 Figure: Typical shape of the risk function s(x)

13 We can use the risk function of a random variable X to determine the probability distribution of this variable. we have the differential after integration we have s(x) = 1 S(x) f (x) = 1 df (x) = 1 S(x) dx S(x) ds(x) dx ds(x) S(x) = s(x)dx x ln S(x) = C 0 s(t)dt, For the initial condition S(0) = 1 we obtain the constant C = 0, S(x) = e x s(t)dt 0.

14 For the distribution function we get F (x) = 1 e x s(t)dt 0, and for the density f (x) = s(x)e x s(t)dt 0.